Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 003, 5 pages      arXiv:0801.1754
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Affine Poisson Groups and WZW Model

Ctirad Klimcík
Institute de mathématiques de Luminy, 163, Avenue de Luminy, 13288 Marseille, France

Received October 31, 2007; Published online January 11, 2008

We give a detailed description of a dynamical system which enjoys a Poisson-Lie symmetry with two non-isomorphic dual groups. The system is obtained by taking the q → ∞ limit of the q-deformed WZW model and the understanding of its symmetry structure results in uncovering an interesting duality of its exchange relations.

Key words: Poisson-Lie symmetry; WZW model.

pdf (178 kb)   ps (137 kb)   tex (10 kb)


  1. Chu M., Goddard P., Halliday I., Olive D., Schwimmer A., Quantization of the Wess-Zumino-Witten model on a circle, Phys. Lett. B 266 (1991), 71-81.
  2. Dazord P., Sondaz D., Groupes de Poisson affines, in Symplectic Geometry, Groupoids, and Integrable Systems (1989, Berkeley, CA), Editors P. Dazord and A. Weinstein, Math. Sci. Res. Inst. Publ., Vol. 20, Springer, New York, 1991, 99-128.
  3. Dorfman I.Ya., Deformations of Hamiltonian structures and integrable systems, Proceedings of Second International Workshop on Nonlinear and Turbulent Processes in Physics (October 19-25, 1983, Kyiv), Editor R.Z. Sagdeev, Harwood Academic Publishers, 1984, Vol. 3, 1313-1318.
  4. Felder G., Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the International Congress of Mathematicians (1994, Zürich), Birkhäuser, Basel, 1995, 1247-1255, hep-th/9407154.
  5. Klimcík C., Quasitriangular WZW model, Rev. Math. Phys. 16 (2004), 679-808, hep-th/0103118.
  6. Klimcík C., q → ∞ limit of the quasitriangular WZW model, J. Nonlinear Math. Phys. 14 (2007), 486-518, math-ph/0611066.
  7. Koszul J.L., Crochet de Schouten-Nijenhuis et cohomologie, Astérisque 137 (1985), 257-271.
  8. Lu J.-H., Multiplicative and affine Poisson structures on Lie groups, Ph.D. Thesis, University of California, Berkeley, 1990, available at
  9. Lukyanov S., Shatashvili S., Free field representation for the classical limit of quantum affine algebra, Phys. Lett. B 298 (1993), 111-115, hep-th/9209130.
  10. Magri F., Morosi C., A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno, Vol. 19, University of Milano, 1984.
  11. Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142.

Previous article   Next article   Contents of Volume 4 (2008)